Bottom Line:
The community structure of complex networks reveals both their organization and hidden relationships among their constituents.This framework is also particularly suitable to monitor the evolution of community structure in temporal networks.An application of consensus clustering to a large citation network of physics papers demonstrates its capability to keep track of the birth, death and diversification of topics.

ABSTRACTThe community structure of complex networks reveals both their organization and hidden relationships among their constituents. Most community detection methods currently available are not deterministic, and their results typically depend on the specific random seeds, initial conditions and tie-break rules adopted for their execution. Consensus clustering is used in data analysis to generate stable results out of a set of partitions delivered by stochastic methods. Here we show that consensus clustering can be combined with any existing method in a self-consistent way, enhancing considerably both the stability and the accuracy of the resulting partitions. This framework is also particularly suitable to monitor the evolution of community structure in temporal networks. An application of consensus clustering to a large citation network of physics papers demonstrates its capability to keep track of the birth, death and diversification of topics.

f1: Effect of consensus clustering on community structure.Schematic illustration of consensus clustering on a graph with two visible clusters, whose vertices are indicated by the squares and circles on the (I) and (II) diagrams. The combination of the partitions (I), (II), (III) and (IV) yields the (weighted) consensus graph illustrated on the right (see Methods). The thickness of each edge is proportional to its weight. In the consensus graph the cluster structure of the original network is more visible: the two communities have become cliques, with “heavy” edges, whereas the connections between them are quite weak. Interestingly, this improvement has been achieved despite the presence of two inaccurate partitions in three clusters (III and IV).

Mentions:
We stress that our goal is not finding a better optimum for the objective function of a given method. Consensus partitions usually do not deliver improved optima. On the other hand, global quality functions, like modularity39, are known to have serious limits404142, and their optimization is often unable to detect clusters in realistic settings, not even when the clusters are loosely connected to each other. In this respect, insisting in finding the absolute optimum of the measure would not be productive. However, if we buy the popular notion of communities as subgraphs with a high internal edge density and a comparatively low external edge density, the task of any method would be easier if we managed to further increase the internal edge density of the subgraphs, enhancing their cohesion, and to further decrease the edge density between the subgraphs, enhancing their separation. Ideally, if we could push this process to the extreme, we would end up with a set of disconnected cliques, which every method would be able to identify, despite its limitations. Consensus clustering induces this type of transformation (Fig. 1) and therefore it mitigates the deficiencies of clustering algorithms, leading to more efficient techniques. The situation in a sense recalls spectral clustering43, where by mapping the original network in a network of points in a Euclidean space, through the eigenvector components of a given matrix (typically the Laplacian), one ends up with a system which is easier to clusterize.

f1: Effect of consensus clustering on community structure.Schematic illustration of consensus clustering on a graph with two visible clusters, whose vertices are indicated by the squares and circles on the (I) and (II) diagrams. The combination of the partitions (I), (II), (III) and (IV) yields the (weighted) consensus graph illustrated on the right (see Methods). The thickness of each edge is proportional to its weight. In the consensus graph the cluster structure of the original network is more visible: the two communities have become cliques, with “heavy” edges, whereas the connections between them are quite weak. Interestingly, this improvement has been achieved despite the presence of two inaccurate partitions in three clusters (III and IV).

Mentions:
We stress that our goal is not finding a better optimum for the objective function of a given method. Consensus partitions usually do not deliver improved optima. On the other hand, global quality functions, like modularity39, are known to have serious limits404142, and their optimization is often unable to detect clusters in realistic settings, not even when the clusters are loosely connected to each other. In this respect, insisting in finding the absolute optimum of the measure would not be productive. However, if we buy the popular notion of communities as subgraphs with a high internal edge density and a comparatively low external edge density, the task of any method would be easier if we managed to further increase the internal edge density of the subgraphs, enhancing their cohesion, and to further decrease the edge density between the subgraphs, enhancing their separation. Ideally, if we could push this process to the extreme, we would end up with a set of disconnected cliques, which every method would be able to identify, despite its limitations. Consensus clustering induces this type of transformation (Fig. 1) and therefore it mitigates the deficiencies of clustering algorithms, leading to more efficient techniques. The situation in a sense recalls spectral clustering43, where by mapping the original network in a network of points in a Euclidean space, through the eigenvector components of a given matrix (typically the Laplacian), one ends up with a system which is easier to clusterize.

Bottom Line:
The community structure of complex networks reveals both their organization and hidden relationships among their constituents.This framework is also particularly suitable to monitor the evolution of community structure in temporal networks.An application of consensus clustering to a large citation network of physics papers demonstrates its capability to keep track of the birth, death and diversification of topics.

ABSTRACTThe community structure of complex networks reveals both their organization and hidden relationships among their constituents. Most community detection methods currently available are not deterministic, and their results typically depend on the specific random seeds, initial conditions and tie-break rules adopted for their execution. Consensus clustering is used in data analysis to generate stable results out of a set of partitions delivered by stochastic methods. Here we show that consensus clustering can be combined with any existing method in a self-consistent way, enhancing considerably both the stability and the accuracy of the resulting partitions. This framework is also particularly suitable to monitor the evolution of community structure in temporal networks. An application of consensus clustering to a large citation network of physics papers demonstrates its capability to keep track of the birth, death and diversification of topics.